There’s something called relative humidity. Air can hold a certain amount of dissolved water. The capacity of the air to hold that water changes with the temperature of the air, with hotter air holding more water than colder air.
If you’re heating up air from the outside, it doesn’t add any humidity to it. While it contains the same amount of water, the relative amount is lower. Because the relative amount is lower, the air actually tries to evaporate moisture out of your skin, since it can hold it better than your skin can. This makes a heated room feel dry.
If you’re cooling down air from the outside, it might actually be holding into more moisture than it can at room temperature. As you cool it down, it actually loses ability to hold water, so it condenses down. This means the air coming out is actually very moist.
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